Lemma 10.78.6. Let R \to S be a flat local homomorphism of local rings. Let M be a finite R-module. Then M is finite projective over R if and only if M \otimes _ R S is finite projective over S.
Proof. By Lemma 10.78.2 being finite projective over a local ring is the same thing as being finite free. Suppose that M \otimes _ R S is a finite free S-module. Pick x_1, \ldots , x_ r \in M whose images in M/\mathfrak m_ RM form a basis over \kappa (\mathfrak m). Then we see that x_1 \otimes 1, \ldots , x_ r \otimes 1 are a basis for M \otimes _ R S. This implies that the map R^{\oplus r} \to M, (a_ i) \mapsto \sum a_ i x_ i becomes an isomorphism after tensoring with S. By faithful flatness of R \to S, see Lemma 10.39.17 we see that it is an isomorphism. \square
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